HW9 - Mathematics Department, UCLA T. Richthammer winter...

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Mathematics Department, UCLA T. Richthammer winter 09, sheet 9 Feb 27, 2009 Homework assignments: Math 170A Probability, Sec. 1 109. Repeat problem 104, now assuming that the arrival time X of the bus is an exponential RV with parameter α = 1 / 15. Answer: (a) P ( X > 10) = R 10 αe - αx dx = e - 10 / 15 0 . 51. (b) P ( X 25 | X > 15) = P ( X > 10) 0 . 51. (memoryless!) 110. Repeat problem 105, now assuming the road has infinite length, and X is exponentially distributed with parameter α . Answer: E ( | X - a | ) = R 0 α | x - a | e - αx dx = R a 0 ( a - x ) αe - αx dx + R a ( x - a ) αe - αx dx = a + 1 α ( e - αa - 1) + 1 α e - αa = a + 2 α e - αa - 1 α = f ( a ), using R ( x - a ) αe - αx dx = - ( x - a ) e - αx - 1 α e - αx (integration by parts). Minimizing f ( a ): f ± ( a ) = 0 gives a = ln 2 α , and we get f ( a ) = ln 2 α . 111.* In the lecture we have argued that under certain assumptions the number of events in an interval [0 , t ] is a Poisson RV with parameter αt by using a discretization argument and the Poisson approximation of binomial RVs. From this we have concluded that the waiting time Y for the first event is an exponential RV with parmeter α . We can also argue directly, using the discretization and a corresponding approximation theorem on the waiting times rather than on the number of successes: We divide the interval [0 , ) into pieces of equal length 1 /n , and let Y n denote the waiting time for the first interval that contains an event. Show the following: (a) The assumptions from 6.3(d) imply that Y n is a geometric RV with parameter α/n . (b) Y n /n Y for large n . (c) Whenever Y n is a geometric RV with parameter p n such that np n α > 0 for n → ∞ , the distribution function of Y n /n converges to the distribution function of an exponential RV with parameter α at every point c . (d) Y has to be an exponential RV with parameter α . Answer: (a) As in 6.3(d) we see that the RVs X i (where X i = 1 if an event occurs in the i -th interval, and X i = 0 otherwise) form a Bernoulli sequence with parameter p = α/n . Y n is the waiting time for the first success in the Bernoulli sequence and thus a geometric RV with parameter p = α/n . (b) If the first success occurs in the i -th interval (which has endpoints i - 1 n and i n ), then Y n = i and Y is in this interval, so | Y - Y n /n | ≤ 1 n . (c) F Y n /n ( c ) = P ( Y n cn ) = P ( Y n c n ), where c n := ± cn ² . We have P ( Y n c n ) = c n k =1 p n (1 - p n ) k = p n 1 - (1 - p n ) cn +1 1 - (1 - p n ) = 1 - (1 - p n ) c n +1 = 1 - [(1 - p n ) n ] c n /n , and in the Poisson approximation theorem we have shown (1 - p n ) n e - α , and we have c n /n = ± cn ² /n c . Thus
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HW9 - Mathematics Department, UCLA T. Richthammer winter...

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